From news.primenet.com!nntp.primenet.com!news1.best.com!sdd.hp.com!swrinde!howland.reston.ans.net!newsfeed.internetmci.com!in2.uu.net!nntp.inet.fi!news.funet.fi!news.cc.tut.fi!news Thu Jun 13 06:51:53 1996 Path: news.primenet.com!nntp.primenet.com!news1.best.com!sdd.hp.com!swrinde!howland.reston.ans.net!newsfeed.internetmci.com!in2.uu.net!nntp.inet.fi!news.funet.fi!news.cc.tut.fi!news From: vnummela@butler.cc.tut.fi (Nummela Ville) Newsgroups: rec.music.makers.guitar.acoustic,rec.music.classical.guitar,rec.music.makers.guitar Subject: a 'FAQ': Physics of Timbre, Strings and Scales Date: 11 Jun 1996 15:34:31 +0300 Organization: TTKK Lines: 311 Sender: vnummela@butler.cc.tut.fi Distribution: world Message-ID: NNTP-Posting-Host: butler.cc.tut.fi X-Newsreader: Gnus v5.1 Xref: news.primenet.com rec.music.makers.guitar.acoustic:39000 rec.music.classical.guitar:19831 rec.music.makers.guitar:97331 a 'FAQ' : Physics of Timbre, Strings and Scales by Ville Nummela (vnummela@cc.tut.fi) 7.6.1996 Updated 10.6.1996 Contents: Introduction 1 ** Sound 2 ** Components of Timbre 3 ** The Non-Ideal World 4 ** 12-Tone Equal Temperament 5 ** Finally In rmmga, one often finds discussions and even debates concerning the tuning of guitar, scales and harmonics. Rather than taking part in the discussion, I wanted to write something more comprehensive for everybody to read and comment on. I by no means consider myself a big 'guru', but here's a few things I've gathered along with playing the guitar and studying physics. This text is intended to be an easy introduction to the subject of sound, timbrae and scales and where these things come from. It assumes little knowledge of maths or physics, but some of the guitar. It took me quite a while to write it so *please* e-mail me if you have any comments / corrections or just wish to say hello! English is not my native tongue, so please bear with any linguistic blunders I may have overlooked. This article is posted to the newsgroups rec.music.makers.guitar.acoustic, rec.music.makers.guitar and rec.music.classical.guitar newsgroups. You may further distribute this text within the internet as long as money is not involved. For any other use, author's permission is required. Thanks to Charles Tauber and Tommi Ilmonen for their comments. 1 ** Sound This may seem a bit theoretical, but I feel I should nevertheless start from the very beginning: What is sound and when does it become music, in terms of the human ear? Everybody knows that sound is microscopic back-and-forth motion of air. This motion is not uniform. All sound is comprised of numerous superimposed vibrations, each of it's own frequency. (With one exception - the sine wave.) The ear does *not* sense the actual motion. Rather, it leaps directly into the 'frequency-domain' representation: Everything is categorized according to the pitch and amplitude. (I shall neglect the finer properties of ear - the sensation of direction, the unique capability to filter unwanted noise out according to it's importance etc.) Even noise, such as your computer's humming, has some spectrum of frequencies. One can think of this as a 'rainbow' of sound. White light is also a superposition of different wavelengths. All the colours are there, though some may be stronger than others. Without the rainbow, the eye would only see white light, but the ear has the inherent ability to spread the spectrum into the small components without any external aid. Sound is often analyzed in terms of the sine wave. It is a continuously oscillating function, easy and beautyful not only for the mathematician, but in terms of everyday physics as well. It is the most simple form of harmonic motion there is. Stick a weight into a spring and set the system into (small) motion and there you have it. The ear too seems to hold it in a special place: The audible 'sound rainbow' of a sine wave is narroved to one thin line, one 'color' or one frequency. No other form of wave does this. Thus, the words 'frequency' and 'sine wave' have become almost inseparable. A certain frecuency is always associated with a certain sine wave and vice versa. 2 ** Components of Timbre Ok, so much for the easy introduction. When does sound become music? An important part of music is rhythm, which is created by percussion and other percussive sounds. That said, I will ignore it for the remainder of the text and only handle individual notes. Most instruments produce sounds that have a certain well-defined pitch. The pitch comes from the (usually) lowest and loudest sine wave / frequency in the sound. This is called the fundamental frequency. There are, however, also additional frequencies present. These are responsible for the characteristic timbre of each instrument. Obviously, the timbrae radically differ from each other. In most cases, the timbre (of any one note) is approximately of the following form: (1)f + 2 a f + 3 b f + 4 c f ....etc. (eq 1) where f is our fundamentan frequency, and the factors a,b,c... are arbitrary constants (actually, everything but constants!) that determine the *relative* amplitudes of the different components, and thus ultimately the timbre. In general, they are smaller than one and diminish as the series progresses. An example: If you damp a string at the 12th fret, the tone changes rather radically. This is because all the 'odd' harmonics (i.e. 1,3,5...) as well as a good deal of other higher harmonics are more or less eliminated by your finger. In terms of (eq 1), the corresponding amplitude coefficients approach zero. This is also the mechanism that allows us to tune the guitar with the harmonics. Damping at the 12th fret, our new fundamental is twice the original, at the 7th it's three times that, at 5th its four etc. Bear this in mind when you get to the tuning part. The most important part of this equation is, however, the whole-number coefficients: 1,2,3 etc. For some reason, our ear seems to favor this kind of frequency combinations. Skipping the lengthy theories, it should be fairly easy to see why the *intervals* (that is, differences between the fundamental notes) we most like are also formed by combining these same whole-number coefficients. Some examples: If f is the fundamental, then an octave would be formed by the notes f and 2f. A (*perfect*) fifth is f and 3/2 f, a fourth 4/3, a major third 5/4, a minor third 6/5. See the pattern? In fact, to this day a good deal of the (more or less traditional) non-western music (eg. Indian classical music) still draws their scales and chord structures from these intervals. And not so long ago, western music did too. The famous Pythagorean system is one version of the story, among many others. A side note: Equation 1 is not the only possibility to produce pleasant-sounding tones. The harmonics' coefficients can be picked other ways too. It just happens to be the way that most real instruments work. I'll let it suffice to say, that timbre and scales are closely connected. With the modern synthesizers it is in principle possible to modify not only the timbre, but it's basic internal structure. Usually you just get noise, but it is still quite possible to get good results. The key lies in understanding, that one has to modify the scales accordingly! (So far I only know of one person that has actually done this. That is, with a scientific basis. Namely William A. Sethares.) 3 ** The Non-Ideal World In practise, real-world instruments, such as the guitar, never obey the (equation 1) exactly. That is, the factors 2,3,4... are only approximately whole numbers. The error usually increases as the numbers grow. I have no confirmation on this, but my own intuition says that in the guitar these coefficients would tend to get warped upwards. (See my reasoning below and please correct me if I'm wrong.) Note that this is not necessarily a bad thing, it may make the sound even more pleasant. [Good examples of instruments that do not obey this 'law' at all would be the brass parts of the drum set, high-hat & others. There, we can't even find any fundamental note to begin with!] Why does this happen? Many things have an effect. I must confess, that I don't know which ones are the most important, but I'd say I can make a good guess. For one, the strings of a guitar resist bending. If we had an infinitely thin or flexible string, then (eq 1) is what we'd get. Also, the strings do not necessarily obey Hooke's law when they are stretched. (= Amount of stretch is directly proportional to the stretching force, or F = -kx (eq 2 - Hooke's law) where k is the string's elastic constant and x the displacement from the midpoint.) These two things mean that our "oscillating system is no longer harmonic". Sorry about the maths. In plain english this means that the frequency is greater than it othervise would be. Climbing our tonal series (of eq 1) upwards, the conditions gradually become less and less ideal -> the *effective* elastic constant k increases with (i.e. as a function of) the number of the overtone (as the bending of the string becomes more important) -> the oscillation frequency increases. Note that k also increases when the string amplitude goes up. The change of pitch is clearly audible when playing loud. Most of us (steel-string acoustic players) find their B strings harder to get in tune than the others. (For electric & classical players this would be the G.) (Part of) the reason for this is related: the mass / stiffness ratios of the adjacent wound and unwound strings are quite different. This makes the strings behave differently in terms of the tonal series. That is, if you were to tune the fundamentals to the same pitch, the upper harmonics would not be in the same places. And following that, the different strings don't want to conform to the same scale. The other (and more important) part is that when you press the string against the fretboard, you also increase the tension of the string and therefore cause the string to vibrate faster. An additional problem is, that different srings have different elastic constants and masses -> the change in frecuency is different. These non-linearities can be compensated for at the saddle. This is what the luthier does when you ask him to fine-tune or set the intonation of the instrument: He simply changes the length of the vibrating string a little bit. Thus, the mid-point of the string need not be exactly on top of the 12th fret. In fact, the vibrating length of the string is always longer tham the theoretical length, moving the mid-point of the string closer to the bridge. Unfortunately, this can't be done perfectly. The timbre and thus the correct amount of compensation depends heavily on the type, condition and even the tuning of the strings. (Which is also why old strings are harder to tune.) So we're bound to keep having problems! Let alone the cheaper or othervise unmaintained guitars, with the (possibly nonexistent) corrections way out of tune! A guitar like this is by definition impossible to get in tune. This problem is also frequently combined with too high action of the strings. Which of course not only makes the playing difficult but further warps the tuning! IMHO, regardless of tone, guitars like this have scared away many more beginners than any other individual factor! 4 ** 12-Tone Equal Temperament So far, I've only talked about whole-number ratios and carefully avoided saying anything about the actual playing situation, or placement of the frets. I hope it doesn't surprise anybody when I 'reveal' that the fretboard indeed does not conform to any whole-number ratio set up. In fact, the distances between the frets, and following suit, the frequency ratios (= our note system) form a 'geometric series'. Or, if you prefer, it is a logarithmic function with a base of two. I'm sorry but here the mathematics just can't be avoided. The whole idea of the equal temperament scale is a mathematical one. It is an arbitrary, out-of-the-hat trick which is only an approximation to our beautyful whole-number world. Of course it does have it's advantages... An equally-tempered scale means, that given a starting point (Say A = 440 Hz), all the other notes' fundamental frequencies can be generated by multiplying or dividing the 'starting point' by a fixed constant. This means that the ratio of any one semitone is exactly equal to every other interval of one semitone. In our familiar 12 - Tone Equal Temperament scale, this constant is the 12th root of 2, or 2^(1/12) = 1.05946... Two is of course the octave and 12 the number of notes within it. Let's assign this number the symbol K. Examples: The perfect fifth is the ratio 3/2. To get to the 12-TET equivalent, one needs to climb 7 notes on the fretboard: K^7 = 1.4983. (=K*K*K*K*K*K*K) Pretty close, eh? There's an error of appr. 0.1 %, I guess I can live with that. Unfortunately, not everything is quite this close. Take a major third: K^4 = 1.2599, as opposed to the ideal value 1.25. Nearly ten times the error! Ouch! [Note that in real world, we hear changes in frequency, rather than in per cent change. Thus the same percentage error is *much* more disturbing at 10 000 Hz than at 100 Hz. The intensity is of course also important.] These errors are of course the main reason behind the difficulties in tuning the guitar with the 5th and 7th fret harmonics. The harmonics produce perfect intervals, while what you really want is the 12-TET interval. (Note that the compensation at the saddle does not affect the harmonics, but the non-ideality of the string does. Old & dirty strings can be way off.) Still, this method can be the most accurate when you have a badly set-up guitar! Personally, I often start with the harmonics and then move on with other methods. No guitar is ever in perfect condition, and the tuning is always more or less a compromise. [Again, other systems have been proposed. For instance, a similar equally tempered system named "Alpha" quite nicely brings both the fifths and the thirds close to their ideal values. The bad news is that we would no longer have octaves!] Now, why do we put up with this then? For one thing, you no longer need to re-fret your axe every time you change the key. It also enables easy transposing (pitch-shifting) of tunes and all the instruments in a band can use it. In the old days, it was very hard for a fretted instrument (like a lute) to play with unfretted ones (like human voice). In short, 12-TET makes a lot of compromises but allows for a far greater flexibility. 5 ** Finally Of course, it all depends on what you want. My guitar duo (or nowadays a trio, we just addded a cello) CaveOss has done some experiments with non-12-TET tunings. In fact, a good deal of our repertoire consists of tunes played in a tuning we call 'Okeli 1'. Most of the stuff we submitted last year to the Tape V project uses it. Check our WWW-page for more info on our 'geometrical tunings'. BTW: CaveOss will be recording again some time this summer. Unfortunately, we've already missed Tape VI! Cheers, Ville # Ville.Nummela@cc.tut.fi http://butler.cc.tut.fi/~vnummela # Guitar duo CaveOss: http://www.hut.fi/~tilmonen/CaveOss.html # Home: Lankiniitynkatu 18, 33580 Tampere Tel. (931) 36 36 312 # Job: Physics / Plasma Lab, room SK116 Tel. (931) (365) 3497 # Tampere University of Technology, Finland. All opinions are mine. -- # Ville.Nummela@cc.tut.fi http://butler.cc.tut.fi/~vnummela # Guitar duo CaveOss: http://www.hut.fi/~tilmonen/CaveOss.html # Home: Lankiniitynkatu 18, 33580 Tampere Tel. (931) 36 36 312 # Job: Physics / Plasma Lab, room SK116 Tel. (931) (365) 3497 # Tampere University of Technology, Finland. All opinions are mine.